Directrix-based ellipses in taxicab geometry

I wrote a couple years ago about conic sections in taxicab geometry, and found that while circles and parabolas seemed to be conic sections, hyperbolas and ellipses didn't. Back in April, Kevin Thompson, who's working on A Compendium of Taxicab Geometry, emailed to point out where the difference lay. While I defined parabolas in terms of a point and directrix, I defined ellipses by the sum of the distances to two foci, and hyperbolas as the difference in distances to two foci. I wasn't aware that there are also directrix-based definitions of ellipses and hyperbolas. In Euclidean geometry, these differing definitions produce identical shapes, but not so in taxicab geometry, where each definition produces a distinct class of shapes. In this post, we'll take a look at directrix-based ellipses in taxicab geometry.

Sum-based ellipses in taxicab geometry are either six- or eight-sided, depending on how their foci are arranged:

These ellipses depend on three factors: a size, and the position of two foci. To base an ellipse on a directrix, we also pick three factors: a line, a focus, and an eccentricity. A point is on the ellipse if the ratio of its distance to the focus to its distance to the directrix is equal to the eccentricity. Such ellipses are four-sided, and come in three varieties.

A vertical or horizontal directrix produces a kite:

When the directrix is 45° to the axes, the ellipse is a trapezoid:

And when the directrix is sloped at something other than 45°, the ellipse becomes an irregular quadrilateral:

Note that, due to their asymmetry, directrix-based ellipses in taxicab geometry only have one focus, not two.

It turns out that only definitions based on directrices produce conic sections. Here's each kind of ellipse created by cutting a taxicab cone (i.e., a pyramid) with a plane:

Thanks for the pointer, Kevin!

In the next post, we'll examine directrix-based hyperbolas.